Journal of Economics and Business

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1 Journal of Economics and Business 66 (2013) Contents lists available at SciVerse ScienceDirect Journal of Economics and Business Liquidity provision in a limit order book without adverse selection Onur Bayar College of Business, University of Texas at San Antonio, United States a r t i c l e i n f o Article history: Received 11 October 2011 Received in revised form 8 January 2013 Accepted 17 January 2013 JEL classification: G12 G14 Keywords: Market microstructure Limit order markets Liquidity Private values a b s t r a c t In this paper, we develop a dynamic model of a limit order market populated with liquidity traders who have only private values. We characterize and analyze the equilibrium order placement strategies of traders and the conditional execution probabilities of limit orders as a function of traders liquidity demand and the state of the limit order book. We solve for the equilibrium of the model numerically, and analyze its properties by performing comparative dynamics analysis. Our analysis shows that changes in the steady state of the limit order book and optimal order placement strategies reflect corresponding changes in the trade-off between order execution risk and the size of potential trading gains. The equilibrium order flow depends on the current state of the limit order book since a trader s optimal trading strategy is largely affected by the time and price priorities of the existing limit orders in the book. We demonstrate how changes in the dispersion of traders private values affect optimal trading strategies and conditional execution probabilities of limit orders. Our main result is that the dispersion in private values across traders has a significant impact on the stationary state of the equilibrium limit order book and the average bid ask spread. A wider distribution of private values leads to more order placement at prices away from the consensus value, and therefore, to a larger bid ask spread. Further, our numerical simulations show that extending the life span of limit orders reduces the average bid ask spread observed in equilibrium. Finally, we find For helpful comments and discussions, I thank Thomas Chemmanur, Robert Taggart, Alan Marcus, Hassan Tehranian, Burton Hollifield, Alex Boulatov, John Wald, Mark Liu, and conference participants at the 2007 FMA meetings. Special thanks to an anonymous referee and to the editor, Ken Kopecky, for helpful suggestions. I am solely responsible for all errors and omissions. Correspondence address: Finance Department, College of Business, University of Texas at San Antonio, San Antonio, TX 78249, United States. Tel.: ; fax: address: onur.bayar@utsa.edu /$ see front matter 2013 Elsevier Inc. All rights reserved.

2 O. Bayar / Journal of Economics and Business 66 (2013) that the equilibrium percentage of market order submissions is also increasing in the dispersion in liquidity traders private values Elsevier Inc. All rights reserved. 1. Introduction In a limit order market, buyers and sellers can submit an order of one of two types. A market order executes immediately the best price posted by previous limit orders. A limit order specifies a particular price for the order and specifies a promise to trade at that price. The limit order book is a list of all unexecuted limit orders. Traders provide liquidity by submitting limit orders and consume liquidity by submitting market orders. Many financial assets are traded in limit order books. There are many stock exchanges around the world where trading takes place completely (e.g., Euronext, Stockholm, Helsinki, Hong Kong, Shanghai, Tokyo, Toronto and various Electronic Communication Networks) or partially through electronic limit order books (NYSE, Nasdaq, London). Despite this prevalence of limit order markets, the theoretical literature on limit order markets is very small. Understanding the dynamic choice between limit orders and market orders is important because rational agents can optimally use different trading strategies depending on the state of the limit order book and their subjective beliefs about the value of financial assets that are traded in these markets. These different strategies, in turn, can generate significant effects on price impact, trading volume, bid ask spreads, and the volatility of market prices. The objective of this paper is to develop a new model of dynamic optimal order placement in a limit order market in order to better understand the economic trade-offs underlying the choice between limit orders and market orders by incorporating the dynamic nature of limit order markets. In our simple setting with symmetrically informed traders each of which has a private valuation of an asset, we focus on the trade-off between the price of an order and its execution probability that is essential to the analysis of traders choice between limit and market orders. This basic trade-off between order price and execution probability can be summarized as follows. A trader can always obtain a larger probability of execution at the cost of a less favorable execution price away from the bid ask spread, which can be interpreted as an implicit cost for demanding liquidity. By definition, a market order is a limit order with execution probability one and therefore has no execution risk at all. The motivation for trade results from agents differences in their private valuations of the asset, which causes the agents to have differences in their incentives to provide or consume liquidity. Traders with more extreme private values are more impatient than traders with moderate private valuations that are close to the mean of the probability distribution of private values. In our model, there is no independently moving common value component and the average private value of traders is constant. Therefore, we abstract from the risk of being picked off (winner s curse). 1 Thus, we focus on the tradeoff between price and execution probability in a limit order market and its effect on liquidity provision in an environment without adverse selection. After modeling the arrival of traders (sellers or buyers) in the market, we characterize and analyze the equilibrium order placement strategies of traders in terms of the state of the limit order book and the execution probabilities of limit orders. We solve for the equilibrium of our model numerically using several parameter specifications and theoretically investigate its properties by performing comparative dynamics analysis. In our model, limit orders last for a finite number of periods, and they cannot be modified or canceled after submission. We devise and implement a numerical algorithm of successive approximations to solve for the stationary Markov equilibrium of the model. The algorithm is based on mapping the liquidity demand/supply of traders into their subjective order execution probabilities. Imposing a monotonicity restriction as in Hollifield, Miller, Såndas, and Slive (2006), we then invert this mapping to derive the liquidity demand/supply of the traders with respect to the execution probabilities of orders at different prices. Using this approach 1 Since a limit order involves a commitment to a price, it is exposed to unfavorable changes in the common value of the asset. This adverse selection risk is called the winner s curse risk or picking-off risk.

3 100 O. Bayar / Journal of Economics and Business 66 (2013) recursively, we find the fixed point of traders liquidity demand/supply and the corresponding execution probabilities in a stationary equilibrium. Our approach lends itself to the characterization of the equilibrium order book in a discrete Markov-chain state representation. Therefore, we are able to analyze the interactions between transient changes in the state of the book and the order flow. Our model yields several interesting new results about the evolution of limit order book in time, the bid ask spread, and the effect of price priority and time priority rules on the optimal placement of limit orders. Our main finding is that the dispersion in private values across traders is a major factor determining the stationary state of the equilibrium limit order book, the bid ask spreads, and the depth of the quotes in the limit order book. When the dispersion of agents private valuations of the asset is small, we predict that submitting limit buy or sell orders at price quotes far from the middle point of the limit order book is less profitable. Since, in this case, agents (on both sides of the book) are more patient, their demand for liquidity is lower, i.e., their tendency to submit more aggressive limit orders and market orders is not strong. Dynamically, this implies that the future execution probabilities of more conservative limit orders submitted (on the other side of the book) in the current time period are lower. Hence, the expected returns to placing buy (sell) limit orders that are far below (above) an investor s own private valuation are lower even though potential gains from limit orders conditional on execution are high. On the other hand, as the dispersion in agents private values increases, the number of impatient traders with higher liquidity demands increases. This makes it more profitable for other traders (with moderate private values) to place more conservative limit orders with larger potential profits, since the execution probability of these orders increases with the presence of more aggressive traders demanding liquidity. Hence, under market conditions with a large dispersion in liquidity traders private valuations, the expected returns to placing buy (sell) limit orders that are far below (above) an investor s own private valuation are higher in equilibrium. Thus, our results show that a wider distribution of private values leads to more order placement at prices away from the consensus value, and therefore, to a larger bid ask spread. The results of our numerical simulations also show that the equilibrium order flow depends on the current state of the limit order book in the sense that an agent s optimal trading strategy is largely affected by the time and price priorities of the existing limit orders in the book. Our model generates another prediction regarding the effect of increasing the life span of limit orders. We find that as the life span of a limit order increases from two to three periods in our model, the average bid ask spread falls in equilibrium. Finally, our results also show that the equilibrium percentage of market order submissions is also increasing in the dispersion in traders private values. The empirical market microstructure literature documents evidence suggesting that traders follow order placement strategies that depend on the state of the market characterized by the limit order book. Using data on limit and market orders from the Paris Bourse, Biais, Hillion, and Spatt (1995) document the persistence of order flow and find that traders react by submitting limit orders in rapid succession when the bid ask spread or the depth at the quotes is large. Hamao and Hasbrouck (1995) study the limit order book of the Tokyo Stock Exchange and also document persistence in order flow. Harris and Hasbrouck (1996) show that the profitability of limit and market orders varies with market conditions on the NYSE. Goldstein and Kavajecz (2000) document substantial shifts in the willingness of traders to place limit orders during extreme market movements in the New York Stock Exchange (NYSE). Sandås (2001) analyzes data from the Stockholm Exchange to test the empirical implications of the static model of Glosten (1994) and rejects them. Hollifield, Miller, and Såndas (2004) show that changes in the relative profitability of limit and market orders are important in explaining the empirical variation in order submission strategies in the Stockholm Stock Exchange. They empirically characterize and estimate optimal order strategies by a monotone function which maps the liquidity demand of the investors into their subjective execution probabilities. They find little evidence against the monotonicity restriction on the estimated trading strategy, which is the basis of our theoretical framework. 2 2 Hollifield et al. (2006) provide another empirical evidence of variation in liquidity supply and demand in the Vancouver Stock Exchange by also incorporating endogenous estimation of arrival rates of traders into their model. They find that traders decision to supply or demand liquidity depends on whether it is scarce or abundant in the current state of the market. Their

4 O. Bayar / Journal of Economics and Business 66 (2013) The trade-offs related to price, execution probability and the risk of winner s curse form the basic framework of the theoretical literature on the choice between limit orders and market orders: see Cohen, Maier, Schwartz, and Whitcomb (1981), Kumar and Seppi (1993), Glosten (1994), Chakravarty and Holden (1995), Handa and Schwartz (1996), Rock (1996), Seppi (1997), Parlour (1998), Foucault (1999), Biais, Martimort, and Rochet (2000), Foucault, Kadan, and Kandel (2005), Wald and Horrigan (2005), Goettler, Parlour, and Rajan (2005, 2009), and Rosu (2009). These papers theoretically analyze prices, trading volumes, and efficiency in limit order markets. Among the theoretical studies listed above, only Parlour (1998), Foucault (1999), Foucault et al. (2005), Goettler et al. (2005), and Rosu (2009) analyze limit order trading in a dynamic setting. Parlour (1998) has a finite-horizon model where traders with private values can place orders either at an ask price A or at a bid price B. Foucault (1999) obtains closed-form solutions for the stationary equilibrium of his infinite-horizon dynamic model, and analyzes the equilibrium implications of the trade-off between price and execution probability, and winner s curse. However, in his model, limit orders expire after only one time period. Goettler et al. (2005) solve numerically for the stationary Markov perfect equilibrium in a dynamic limit order market. Their focus is on transaction costs, picking-off risk due to adverse selection, and the relationship between the consensus (fundamental) value of an asset and the characteristics of the limit order book. Unlike in the last two studies, there is no risk of limit orders being picked off (adverse selection risk) in our model, since the common value component is fixed. Therefore, the main contribution of our paper to the literature on limit order markets is to study liquidity provision in an environment without adverse selection. 3 The paper is organized as follows. We outline our model in Section 2. In Section 3, we solve for the equilibrium of our model numerically and describe our solution algorithm. Next, we present an illustrative numerical example in Section 4. In Section 5, we present and discuss the results of our analysis of comparative dynamics. We describe the empirical implications of our model in Section 6. Section 7 concludes. 2. Model This section presents the theoretical model we analyze in detail in later sections. First, we provide assumptions on the trading rules and trader preferences Description of the dynamic trading game We consider the market for a single risky asset. Traders with different private valuations for the asset arrive sequentially in the market with an opportunity to trade. Agents can place an order to buy or sell one unit of the asset at a price chosen from the finite set P {p 1,..., p N }, (1) where p i < p i+1 for any i {1,..., N 1}. In our numerical simulations, we specifically set N = 4, with p 1 = 30, p 2 = 31, p 3 = 32, p 4 = The variable t refers to both the time period t when the order is submitted and to the agent whose turn it is to place an order at time t, where t {0, 1,... }. Upon arriving at the market, if the trader t decides to place an order for one unit of the asset, she determines the type of her order. Once an order has been submitted, it will either trade immediately evidence suggests that if liquidity is scarce and therefore valuable, liquidity traders supply liquidity, but when its abundantly available, they prefer to demand it. 3 Foucault et al. (2005) also consider a dynamic model of a limit order market motivated by traders who have differences in waiting costs, and they analyze bid ask spread dynamics, market resiliency, effect of tick size, and time to execution. They require limit order traders to undercut existing orders, without the option to submit orders at or away from the quotes. Rosu (2009) presents a continuous time version of the Foucault et al. (2005) model, and endogenizes their undercutting result. Goettler et al. (2009) and Rosu (2010) extend their previous models to introduce asymmetric information. 4 It is possible to model the price grid P to include more than 4 consecutive prices. For simplicity, we focus here on specifications with N = 4 in our numerical simulations.

5 102 O. Bayar / Journal of Economics and Business 66 (2013) (i.e., it is a market order) or enter the queue of unexecuted orders which is referred to as the limit order book. If a limit order is not executed in M periods after it has been submitted, it automatically expires at the end of the Mth period. In our model, we first analyze model specifications with M = 2. Later, we also model cases with M = 3, and analyze the effect of increasing M (from 2 to 3) to the limit order market equilibrium in our model. We assume that once a limit order is submitted, it cannot be canceled. At any time t, the limit order book consists of outstanding orders to buy and sell stock at some feasible prices. Note that the maximum number of outstanding limit orders at any given time t and the maximum depth at any given price quote p i are both equal to M. The number of outstanding limit orders at given prices and the age of each order completely determines the state of the order book. An implication of sequential trading is that none of the existing limit orders can have the same age in a given state of the market. Given that limit orders can last up to M periods and the price set P is finite, it follows that the number of possible states of the limit order book is also finite. In our basic model with four price points (N = 4) where the limit orders last for M = 2 periods, the total number of states S is equal to 61. The state space is denoted by and each unique state of the limit order book is denoted by ω s. For example, ω 51 = {+2, 0, 1, 0}, (2) corresponds to the state of the limit order book at time t when there is already a limit buy order at price p 1 submitted at time t 2 and a limit sell order submitted at price p 3 at time t 1. Suppose that this is the state of the market at time t and consider the states to which the order book can make a transition at time t + 1 depending on the order strategy of the trader at time t. 1. Do not place an order or place a market sell order at price p 1 : ω 16 = {0, 0, 2, 0} 2. Place a limit sell order at p 2 : ω 41 = {0, 1, 2, 0} 3. Place a limit sell order at p 3 5 : ω 24 = {0, 0, ( 1, 2), 0} 4. Place a limit sell order at p 4 : ω 61 = {0, 0, 2, 1} 5. Place a limit buy order at p 1 : ω 33 = { +1, 0, 2, 0} 6. Place a limit buy order at p 2 : ω 29 = {0, + 1, 2, 0} 7. Place a limit buy order at p 3 : ω 1 = {0, 0, 0, 0} We can further explain the construction of the state space of the limit order book as follows. With M = 2, there can be one of 7 possible order combinations at any given price quote p i : (1) no orders denoted by 0; (2) one buy order with age 1 denoted by +1; (3) one sell order with age 1 denoted by 1; (4) one buy order with age 2 denoted by +2; (5) one sell order with age 2 denoted by 2; (6) two buy orders with ages 1 and 2 respectively, denoted by (+1,+2); (7) two sell orders with ages 1 and 2 respectively, denoted by ( 1, 2). We first start with the empty book state, which is denoted by ω 1. We then consider all states where there exist limit orders at only one price quote: out of the above 7 possible order combinations, 6 of them actually contain orders and each of them can be placed at N = 4 different prices, which yields 24 states from ω 2 to ω 25. Next, we consider all states where there exist limit orders at two different price quotes: out of the 4 singleorder combinations above (i.e., 2, 3, 4, and 5), there are 12 (4 3) possible two-way permutations of which only 6 are feasible in a limit order book: { + 1, + 2}, { + 1, 2}, { 1, 2}, { + 2, + 1}, { + 2, 1}, and { 2, 1}. Given a price grid with N = 4 units, each of these 6 two-way permutations can be placed into the limit order book in 6 (two-way combinations of 4 prices) different ways, which yields 36 other states from ω 26 to ω 61. In general, for any N 4 and M = 2, the number of states is given by 6 : S = 1 + 6N + 6 N(N 1). (3) 2 5 The notation ( 1, 2) indicates that there are two limit sell orders outstanding at the same price quote with ages 1 and 2 respectively. For example, in state ω 24, there are two limit sell orders outstanding at price p 3. Similarly, to denote two limit buy orders outstanding at the same price quote, we use the notation (+ 1, + 2). 6 For example, if M = 2 and N = 5, the number of states is given by S = = 91.

6 O. Bayar / Journal of Economics and Business 66 (2013) Similarly, for any N 4 and M = 3, the number of states is given by 7 : S = N + 36 N(N 1) N(N 1)(N 2). (4) 6 We will refer to the state of the market at time t as s t. Given the structure of the model, in certain states of the market, a trader can outbid the best quote, bid at the best quote, or underbid the best quote. The ask price is the lowest quoted price of existing limit sell orders at time t, and it is denoted a t. If the sell side of the book is empty, a t =+. Similarly, the bid price, b t, is the highest limit price of existing limit orders to buy at time t. If the buy side of the book is empty, b t =. To denote the state dependence of bid and ask prices, we also use the notation a(s t ) and b(s t ) for the ask and the bid, respectively. Let K and L be such that p K = max {b(s t ), p 1 } and p L = min {a(s t ), p N }. The trader s decision in state s t will be denoted with the decision variables d s k (t), d b l (t) for k = K,..., N and l = 1,..., L. If d s k = 1 for some k where p k > b(s t ), then the trader submits a limit sell order at price p k. If the trader places a market sell order at the bid price b(s t ) = p K, such that d s K (t) = 1, then the order is immediately matched with the oldest outstanding limit order at the bid price b(s t ). If d b l (t) = 1 for some l where p l < a(s t ), the trader submits a limit buy order at price p l. If the trader places a market buy order at the ask price a(s t ) = p L, such that d b L (t) = 1. If the trader does not submit any order, then d s k (t) = 0, d b l (t) = 0 for all k and l. As implied by these definitions, orders are first prioritized by price and then by submission time Trader preferences We assume that the traders are symmetrically informed. The rationale for trading results from the different liquidity demands of the agents characterized by their private valuations of the asset. Depending on their private valuations, agents may want to supply or demand liquidity to the market or not to submit an order at all. This renders the market as a private value auction. Hence, the decision to trade and the choice of the type and the price of an order is endogenous. The tth agent s private valuation for the asset is denoted u t. We consider an i.i.d. uniform distribution for private values which is centered at the mid-point of the prices at which orders can be submitted. Thus, u t is distributed independently and identically across agents uniformly with support [A, B], where A = p 1 w, B = p N + w, and w is a positive constant. The private valuation u t can be interpreted as the tth agent s preference for liquidity and captures her willingness to hold the asset. Even though there is no explicit common value component in this setting, we can think of the common value of the asset as being fixed at (A + B)/2, or equivalently at the middle of the price grid P, which is equal to (p 1 + p N )/2. 8 Thus, in our limit order market setting without adverse selection (the risk of being picked off), the only motivation for trade is agents differences in their private valuations of the asset. All agents are assumed to be risk neutral and maximize their expected utility. Conditional on arriving in the market at state s t, the expected payoff of a trader who submits an order crucially 7 In this case, there are a total of 15 possible order combinations at a single price quote: (1) no orders denoted by 0; (2) one buy order with age 1 denoted by +1; (3) one sell order with age 1 denoted by 1; (4) one buy order with age 2 denoted by +2; (5) one sell order with age 2 denoted by 2; (6) one buy order with age 3 denoted by +3; (7) one sell order with age 3 denoted by 3, (8) two buy orders with ages 1 and 2 respectively, denoted by (+1,+2); (9) two sell orders with ages 1 and 2 respectively, denoted by ( 1, 2); (10) two buy orders with ages 1 and 3 respectively, denoted by (+1,+3); (11) two sell orders with ages 1 and 3 respectively, denoted by ( 1, 3); (12) two buy orders with ages 2 and 3 respectively, denoted by (+2,+3); (13) two sell orders with ages 2 and 3 respectively, denoted by ( 2, 3); (14) three buy orders with ages 1, 2, and 3 respectively, denoted by (+1,+2,+3); (15) three sell orders with ages 1, 2, and 3 respectively, denoted by ( 1, 2, 3). Then, the first state is the empty limit order book. The number of all states where there exist limit orders at only one price quote is 14N. The number of all states where there exist limit orders at two different price quotes is 36N(N 1)/2. Finally, the number of all states where there exist limit orders at three different price quotes is 24N(N 1)(N 2)/6. 8 Note that it is only a semantic distinction to state that there is no common value and private values are uniformly distributed over [A, B] or to state that the common value is fixed at (A + B)/2 and private values are uniformly distributed over [ (B A)/2, + (B A)/2]. We thank to an anonymous referee for suggesting this insightful interpretation of our model.

7 104 O. Bayar / Journal of Economics and Business 66 (2013) depends on the conditional execution probability of that order. Conditional execution probabilities for buy and sell orders at each price p i at time t, in state s t are denoted by b i (s t) and s i (s t), respectively. We will show below how these probabilities can be computed given a rule that monotonically maps a trader s liquidity demand into those execution probabilities. Suppose that a trader with valuation u submits a buy order at price p i. Then her expected payoff conditional on the information at time t is equal to b i (s t)[u p i ]. Similarly, the conditional expected payoff of a trader with valuation u is equal to s i (s t)[p i u] when she submits a sell order at price p i. The trader chooses d s k {0, 1} for k = K,..., N and d b {0, 1} for l = 1,..., L to maximize l N d s k s k (s t)[p k u] + k=k subject to the constraint: N L d s k + d b 1. l k=k l=1 L l=1 d b l b l (s t)[u p l ], (5) Indifference threshold valuations To find the optimal decision rule that maps a trader s private valuation u t to order submission prices in state s t, we first define some threshold valuations. For i > 1, the parameter b i (s t) denotes the threshold valuation that makes a trader just indifferent between submitting a buy order at price p i and submitting a buy order at price p i 1 in state s t. Thus, for all i > 1 such that p i a(s t ), a trader strictly prefers submitting a buy order at p i to submitting a buy order at p i 1 if and only if trader t s private valuation u t is greater than the threshold b i (s t). Similarly, for i < N, s i (s t) denotes the threshold valuation that makes a trader just indifferent between submitting a sell order at price p i and submitting a sell order at price p i+1 in state s t. The parameters s N (s t) and b 1 (s t) are the limits of the closed interval [ s N (s t), b 1 (s t)], in which a trader makes a no-order decision. The following lemma builds an important link between conditional order execution probabilities { b i (s t), s i (s t)} N i=1 and traders private valuations: Lemma 1. For two buyers with valuations u and u, u > u, who optimally choose to submit buy orders at prices p b and p b i j, respectively, we have: (u u )( b b i j ) 0. (6) Similarly, for two sellers with valuations u and u, u > u, who optimally choose to submit sell orders at prices p s i and p s j, respectively, we have (u u)( s s i j ) 0. Proof. By optimality, b i (u p b i ) b j (u p b j ), b j (u p b j ) b i (u p b i ). Multiplying the second inequality by 1 and adding and rearranging gives: (u u )( b b i j ) 0. The proof is symmetric for the sell side. Lemma 1, adapted into our setting from Hollifield et al. (2006), shows that execution probabilities of optimally placed orders must be monotone with respect to the traders private valuation, which measures their liquidity demand. This means that the higher the private valuation of a limit order buyer

8 O. Bayar / Journal of Economics and Business 66 (2013) (u > u), the higher will be the execution probability of her limit buy order ( b j b i ) since buyers with higher private values have a greater liquidity demand on the buy side. In fact, if a buying trader s private valuation exceeds some threshold, she will submit a market buy order with probability 1. Similarly, the lower the private valuation of a seller (u < u ), the higher will be the execution probability of her limit sell order ( s i s j ) since sellers with lower private values will be more impatient to sell the asset and therefore have a greater liquidity demand on the sell side. Further, if a selling trader s private valuation is below some threshold, she will submit a market sell order with probability 1. Thus, according to Lemma 1, optimality requires that the mapping from valuations to execution probabilities is monotone. Traders with extremely low or high values of u t have a higher willingness to trade the asset immediately, whereas traders with moderate values of u t demand liquidity more patiently and only if the current state of the order book presents them profitable trading opportunities. The next proposition shows that, as the limit order price moves away from the bid ask spread, the conditional order execution probability decreases monotonically. and p b j Proposition 1. Suppose that there exist two agents who optimally submit buy orders at prices p b i respectively, and b b i j. Then, it follows that p b p b i j. Similarly, if p s i and p s j are some optimal prices to sell and s i s j, it follows that p s i p s j. Proof. Suppose that p b > p b i j. Since it is optimal for some agent to submit a buy order at p b i, optimality requires that there exists a u such that u > p b > p b i j. Then for any such u, we have b j (u p b j ) > b i (u p b i ), which implies that the agent with private value u will prefer to submit a buy order at p b j p b i, hence a contradiction. The proof is symmetric for the sell side. (7) rather than Together with Lemma 1, Proposition 1 implies the monotonicity of the mapping from private valuations to order prices that could be optimal for some trader given the state of the limit order book. From Lemma 1, we know that for two buyers with valuations u and u (where u > u), who optimally choose to submit buy orders at prices p b and p b respectively, it follows that b b i j i j. Further, Proposition 1 shows that in this case, the price p b at which the trader with the higher private valuation u submits j his optimal buy order is not less than the price p b at which the buying trader with the lower private valuation u submits her optimal buy order, i.e., p b p b j i. Given the monotonicity of optimal order i submissions in trader valuations, the optimal order placement strategy is fully characterized by the following proposition, which is also adapted to our model from Hollifield et al. (2006). Proposition 2 (Optimal order placement strategy). Suppose that a trader with private valuation u arrives at the market at time t and the state of the market is s t. Trading opportunities in the limit order book are characterized by the conditional execution probabilities s k (s t) and b l (s t) for sell order choices k = K,..., N and buy order choices l = 1,..., L, respectively. The set of prices that could be optimal for some trader given the state of the book are p s K < < p s N on the sell side, where p s K b(s t ), with p b L > > p b 1 defined similarly for the buy side, where p L a(s t ). Defining s k (s t) = p k (p k+1 p k ) s k+1 (s t) s k (s t) s k+1 (s t), k = K,..., N 1; (8) b l (s t) = p l + (p l p l 1 ) b l 1 (s t) b l 1 (s t) b l (s t), l = 2,..., L; (9) b 1 (s t) = s N (s t) = b 1 (s t)p 1 + s N (s t)p N b 1 (s t) + s N (s. (10) t)

9 106 O. Bayar / Journal of Economics and Business 66 (2013) The optimal decision rule is given by { 1, u s d s K (s K (s t) t) =, d s k (s t) = d b l (s t ) = d b L (s t) = 0, otherwise { 1, u ( s k 1 (s t), s k (s t)] 0, otherwise { 1, u ( b l (s t), b l+1 (s t)] 0, otherwise { 1, u > b L (s t). 0, otherwise (11), k = K + 1,..., N; (12), l = 1,..., L 1; (13) The threshold valuations s k (s t) and b l (s t) given in Eqs. (8) and (9) respectively are obtained by solving for private valuations that make a trader just indifferent between submitting an order at two adjacent prices: s k (s t)(p k s k (s t)) = s k+1 (s t)(p k+1 s k (s t)), (15) b l (s t)( b l (s t) p l ) = b l 1 (s t)( b l (s t) p l 1 ). (16) Lemma 1 and Proposition 1 together imply that traders optimal order placement strategies are monotone in the following sense. For instance, when a t = p L and all feasible buy order prices p L > > p 1 are optimal for some buyers, then the corresponding threshold valuations form a monotone sequence b L (s t) > > b 1 (s t) that divides the valuation line into L + 1 intervals. The optimal decision rule maps these valuation intervals into bidding strategies at L different prices. Thus, traders with higher valuations submit buy orders with higher prices, which have a higher execution probability. Symmetric arguments apply to the sell side. Hence, traders with lower valuations submit sell orders with lower prices, which have a higher execution probability. For a buyer indifferent between submitting a buy order at the lowest possible price p 1 and not entering an order, the threshold value b 1 (s t) solves: (14) b 1 (s t)( b 1 (s t) p 1 ) = 0. (17) Thus, b 1 (s t) = p 1. Similarly, for the sell side, the indifference equation s N (s t)(p N s N (s t)) = 0 implies that s N (s t) = p N. But then, [ s N, b 1 ] = [p N, p 1 ] is an empty set (since p N > p 1 by assumption), which shows that in the absence of a time-varying common value of the asset or trading cost differentials, the no-order decision is dominated in every state of the market for all agents. 9 In this case, if selling at price p N and buying at price p 1 are not dominated, s N is identically equal to b 1, and therefore, denotes the valuation at which the trader is indifferent between submitting a sell order at price p N and a buy order at price p 1. This follows from the following indifference equation: b 1 (s t)( s N (s t) p 1 ) = s N (s t)(p N s N (s t)). (18) A representation of a trader s optimal order placement strategy as a monotone function of her private valuation is shown in a graphical example depicted in Fig Solving for the equilibrium In this section, we explain the numerical solution method that we implement to solve for the stationary Markov equilibrium of our dynamic limit order trading game. 9 In our setting, an order placement strategy is dominated if there exists no agent with some private value u [A, B] such that it is optimal for that agent to follow that strategy.

10 O. Bayar / Journal of Economics and Business 66 (2013) A θ1 s θ2 s θ3 s θ4 s = θ1 b θ2 b θ3 b θb 4 B Sell at p 1 Sell at p 2 Sell at p 3 Sell at p 4 Buy at p 1 Buy at p 2 Buy at p 3 Buy at p 4 Fig. 1. Optimal order placement strategy as a function of the trader s private valuation. This figure presents a visualization of the optimal order strategy of a trader arriving at time t as a function of her private valuation of the asset. The sell threshold valuations s k (st) for k = K,..., N and the buy threshold valuations b l (st) for l = 1,..., L form a monotone sequence, which partitions the support [A, B] of the private value distribution into the trader s optimal order choices. In this example, the price grid has N = 4 units, and the limit order book is in a state s t, where K = 1 and L = Algorithm The algorithm we implement to solve for the equilibrium is based on traders optimal order placement strategy derived in Proposition 2 and the link between optimal order choice rules and the conditional execution probabilities of submitted orders. It solves for the fixed point of the correspondence on the space of optimal order placement rules defined by the monotone sequence of threshold valuations we explained in the previous section. Let be the space of optimal order placement decision rules and ( (s t )) = { b i (s t), s j (s t); i = 1,..., L st ; j = K st,..., N} be an arbitrary element of implied by the conditional execution probabilities (s t ) according to Proposition 2. Note that ( (s t )) is a mapping defined on the trading opportunities (s t ) = { b i (s t), s i (s t); i = 1,..., L st ; j = K st,..., N} available in state s t at time t. But once an optimal decision rule ( (s t )) is specified, it also directly affects the conditional execution probabilities of the submitted orders. In other words, the trading opportunities defined by the conditional execution probabilities in a given state are also a function of the specific decision rule implemented and therefore, we can denote them as ((s t )). Thus, in a stationary Markov equilibrium of our model, the sequence of optimal order placement rules n ( n (s t )) and the trading opportunities n ( n 1 (s t )) will converge to their stationary limits ( (s t )) and ( (s t )) respectively. To understand the derivation of conditional order execution probabilities implied by an optimal decision rule, one should first note that an optimal order placement rule (s t ) determines the probabilities of all feasible order submissions in any given state s t. Suppose that a trader t arrives in the market in a state s t such that the sell side of the book is not empty, i.e., a(s t )< +. Using the optimal order placement strategy given in Proposition 2 and recalling that each trader s private valuation is uniformly distributed over the interval [A, B] with cumulative distribution function F, the conditional probability that we will observe a market buy order is then equal to P(d b L (s t) = 1) = P(u > b L (s t)) = 1 F( b L (s t)). (19) The conditional probability that a limit buy order is submitted at time t at a particular price p l, l = 1,..., L 1 is equal to P(d b (s l t ) = 1) = P( b l (s t) < u b l+1 (s t)) = F( b l+1 (s t)) F( b l (s t)). (20) One can also obtain the conditional order submission probabilities for all feasible sell orders in state s t similarly. 10 Thus, an optimal trading strategy monotonically maps traders private valuations into their optimal order submissions and therefore, into conditional order submission probabilities across feasible price quotes in any state s t. Second, one should also note that since there are finitely many states of the limit order book, the transition possibilities among these states are well defined and finite. Given an order placement rule 10 Recall that once an optimal trading strategy is specified, the no-order decision is dominated in our setting without adverse selection. Therefore, the probability of no-order submission is zero.

11 108 O. Bayar / Journal of Economics and Business 66 (2013) , conditional order submission probabilities obtained in Eqs. (19) and (20) allow us to compute the one-period state transition probabilities between any two states in. If a stationary equilibrium exists, we know that will also be the optimal decision rule of the traders arriving at times t + 1 through t + M, since the underlying probability distribution of their private valuations is independently and identically distributed as that of the trader arriving at t. Thus, given that the decision rule also maps the private valuations of the traders in the following periods t + 1 through t + M to their optimal trading strategies and that limit orders last for M periods, we can then compute the conditional execution probabilities of all feasible limit orders that can be submitted at time t in any state s t. We will illustrate this point via a numerical example (where M = 2) in the next section. Initially, the algorithm starts with an arbitrary decision rule or order placement strategy o (s t ) = { b,o (s i t ), s,o (s i t ), i = 1,..., N} st. This initial trading strategy consists of a monotone sequence of indifference valuations and the associated decision rule defined in Proposition We assume that initially, none of the feasible sell order prices p s K < < p s N and feasible buy order prices p b L > > p b 1 are dominated at the very beginning of the algorithm, i.e., for any feasible buy or sell order price in any state s t, there exists a trader with private value u for whom it is optimal to place that order in state s t. 12 The next step of the algorithm is to determine the conditional execution probability of any feasible limit order at any given state, 1 (s t ) = { b,1 (s i t ), s,1 (s j t ), i = 1,..., L st ; j = K st,..., N} given the initial trading strategy o (s t ). Thus, in general, at the nth iteration of the algorithm, this step involves the determination of the trading opportunities n (s t, n 1 (s t )) (in any state s t ) implied by the order placement strategies n 1 (s t ) derived at the previous (n 1)th iteration. After finding 1 (s t ) ( n (s t )) for all s t, the next step is to determine the optimal trading strategy 1 (s t ) ( n (s t )) for all s t as described in Proposition 2. This step of the algorithm requires in many cases (states of the market) the iterated elimination of some dominated order choices which the trader would never find optimal in state s t regardless of her private valuation u. This process of iteratively eliminating the dominated order choices reduces the set of feasible buy and sell order choices in a state s t to a set of price quotes and their associated order types, which an agent can find optimal in that state of the market given her private valuation u and the trading opportunities n (s t ). These steps constitute a single iteration of the algorithm. They are then iterated until the optimal trading strategy n (s t ) and the trading opportunities n (s t ) converge to some stationary limit points (s t ) and (s t ), respectively, for all states s t. 4. An illustrative example Suppose that the private valuation u of any trader t is drawn independently from the uniform distribution with support [29, 34], and the price set (with N = 4) is equal to P = {p 1 = 30, p 2 = 31, p 3 = 32, p 4 = 33}. Each unexecuted limit order can last up to M = 2 periods. Assume that upon arriving in the market at time t, the trader t finds the limit order book empty, i.e., s t = ω 1 = {0, 0, 0, 0}. Given the underlying probability distribution of a trader s private valuation and rationally expecting that any trader will use the same initial decision rule o (s t ) given by { s,o = 29.5, s,o = 30, s,o = 30.5, s,o = 31, b,o = 32, b,o = 32.5, b,o = 33, b,o = 33.5} The initial trading strategy o (s t) is not necessarily a member of the space of the optimal order placement decision rules. In other words, different from the construction in Proposition 2, o (s t) is not derived from the available trading opportunities in the market characterized by a set of conditional execution probabilities s,o (s k t), b,o (s l t). Although we denote this initial sequence o (s t) as a function of the state s t, it can be state independent as well, i.e., o (s t) = o for all s t. In fact, we first tested our algorithm by using various state independent rules and then switched to randomized state dependent initial rules in order to test the robustness of our convergence results. We verified that the final results obtained after the algorithm converges to the steady state do not depend on the initial trading strategy. 12 At the first iteration, one can also specify a no-order region, where b,o (st) > s,o 1 N (st), although it follows from Proposition 2 that the no-order strategy is dominated in our setting with no winner s curse risk. Therefore, no-order placement is not a part of optimal trading strategies in subsequent iterations.

12 O. Bayar / Journal of Economics and Business 66 (2013) for all s t, she can infer the conditional execution probabilities of all feasible orders she could submit in each state s t, which are denoted by 1 ( o (s t )) = { b,1 (s i t ), s,1 (s j t )}, where i = 1,..., L st and j = K st,..., 4. We will illustrate this simple computation only for b,1 (s 3 t ), which is the conditional execution probability of a limit buy order submitted at price p 3 = 32, when the limit order book is in state ω 1 = {0, 0, 0, 0}. After submitting this order at time t, the limit order book will be in the following state at time t + 1: s t+1 = ω 4 = {0, 0, +1, 0}. (21) Then, the probability that this limit buy order is executed at time t + 1 (matched by a market sell order submitted at p 3 ) is equal to F( s,o 3 ) = F(30.5) = ( )/(34 29) = 0.3, given that trader t + 1 will use the trading strategy o. Then, with probability 0.7, the outstanding limit order at price p 3 will not be executed at time t + 1. Consequently, it can be executed at time t + 2 only in the following cases 13 : 1. Trader t + 1 submits a limit buy order at p 1 with probability F( b,o 2 ) F( b,o 1 ) = ( )/(34 29) = 0.1 and s t+2 = ω 27 = { +1, 0, +2, 0}. 2. Trader t + 1 submits a limit buy order at p 2 with probability F( b,o 3 ) F( b,o 2 ) = ( )/(34 29) = 0.1 and s t+2 = ω 29 = {0, + 1, +2, 0}. 3. Trader t + 1 submits a limit buy order at p 3 with probability F( b,o 4 ) F( b,o 3 ) = ( )/(34 29) = 0.1 and s t+2 = ω 20 = {0, 0, (+ 1, +2), 0}. 4. Trader t + 1 submits a limit sell order at p 4 with probability F( s,o 4 ) F( s,o 3 ) = ( )/(34 29) = 0.1 and s t+2 = ω 55 = {0, 0, +2, 1}. 5. Trader t + 1 does not submit any order with probability F( b,o 1 ) F( s,o 4 ) = (32 31)/(34 29) = 0.2 and s t+2 = ω 12 = {0, 0, +2, 0}. In all these five cases, the bid price b(s t+2 ) is equal to p 3 at time t + 2, and the conditional probability of execution at time t + 2 is therefore equal to the probability that trader t + 2 submits a market sell order at p 3, which is equal to F( s,o 3 ) = F(30.5) = 0.3. Thus, given the decision rule o and by backward induction, the execution probability of a limit buy order submitted at price p 3 at time t in state s t = ω 1 = {0, 0, 0, 0} is equal to b,1 (s 3 t ) = P(execution at time t + 1) + P(execution at time t + 2) = ( ) = (22) The same logic can be applied to find the conditional execution probabilities n of any feasible order in any state implied by the agents trading strategy n Next we consider an example for the next step of the algorithm that maps the agent s private valuation u into her optimal trading strategy n ( n (s t )) given the trading opportunities n (s t ) available in state s t. Now, we assume that upon arriving in the market at time t, the trader t finds the book in state ω 51, i.e., ω 51 = { +2, 0, 1, 0}, where a(ω 51 ) = p 3 and b(ω 51 ) = p 1. Then, the trader infers the trading opportunities s,1 (s t ) = {1.00, 0.48, 0.16, 0.07} and b,1 (s t ) = {0.18, 0.36, 1.00, 0} as shown above. Note that in state ω 51, the agent cannot submit a buy order at p 4 because the ask price is equal to p 3, so we set b,1 = 0. 4 Given the conditional execution probabilities 1 (s t ) in state ω 51, our algorithm initially assumes that all feasible sell order choices at prices p s 1,..., p s 4 and buy order choices at prices p b 1,..., p b 3 can be optimal for some traders, and applies the decision rules given in Proposition 2 to find a 13 Trader t + 1 can also submit a limit buy order at p 4. But if this occurs, the existing limit buy order at p 3 (submitted at time t) has zero probability of execution at time t + 2 due to the price priority rule. 14 This line of analysis can be also generalized to the case where limit orders last for M periods where M > 2. In this paper, we also analyze the case when M = 3.

13 110 O. Bayar / Journal of Economics and Business 66 (2013) monotone sequence of threshold valuations 1 ( 1 (s t )) that characterizes the optimal trading strategy of the trader as a function of her private value. This yields the following sequence in our example: { s,1 1 = 29.08, s,1 2 = 30.50, s,1 3 = 31.22, s,1 4 = 30.84, b,1 1 = 30.84, b,1 2 = 32.00, b,1 3 = 32.56} However, it follows that given the trading opportunities 1 (ω 51 ), the region ( s,1 3 (ω 51), s,1 4 (ω 51)] = (31.22, 30.84] is not well defined, and submitting a limit sell order at price p 4 in state ω 51 is dominated by submitting a limit sell order at p 3 or a limit buy order at p 1. In such cases, where the algorithm finds dominated order choices, it iteratively eliminates them from consideration until the resulting final sequence of indifference valuations is monotone. In this case, for instance, the algorithm determines the private valuation u that makes the trader indifferent between submitting a limit sell order at price p 3 and a limit buy order at price p 1. u = b,1 1 (ω 51 )p 1 + s,1 3 (ω 51)p 3 b,1 1 (ω 51 ) + s,1 3 (ω 51) = = (23) Setting s,1 3 = s,1 4 = b,1 1 = u = 30.94, the algorithm determines the optimal trading strategy 15 1 ( 1 (ω 51 )) = {29.08, 30.50, 30.94, 30.94, 30.94, 32.00, 32.56}. (24) After finding the optimal trading strategy 1 ( 1 (s t )) for all states s t in this fashion, the first iteration of the algorithm is completed. These iterations continue until we obtain convergence for the optimal strategies n (s t ) and conditional execution probabilities n (s t ) respectively. The convergence limits (s t ) and (s t ) for all s t characterize the symmetric stationary equilibrium of our dynamic limit order trading game. In our specific example, the conditional execution probabilities and optimal strategies for s t = ω 51 converge to s, (ω 51 ) = {1.00, , , 0}, b, (ω 51 ) = {0, , 1.00, 0}, (ω 51 ) = {29, , , , , , }. (25) Thus, we can characterize the optimal trading strategy (ω 51 ) in state ω 51 = { +2, 0, 1, 0} as follows. It is never optimal for a trader to submit sell orders at prices p 1 = 30 and p 4 = 33 and buy orders at price p 1 = If the trader s private value u is in the interval [29, ], she will submit a limit sell order at p 2 = 31 with an execution probability of If u ( , ], she will submit a limit sell order at p 3 = 32 with an execution probability of If u ( , ], she will submit a limit buy order at p 2 = 31 with an execution probability of Finally, if u ( , 34], she will submit a market buy order at p 3 = Note that although placing a limit sell order at p 4 is dominated, we still set s,1 = for notational and computational 4 convenience and it does not affect the following iterations of the algorithm. 16 Consider why it is never optimal to submit a market sell order at p 1 = 30. Note that since the support of the probability distribution for private value u is [29, 34], the largest net payoff from selling at p 1 = 30 will be realized by a trader with the minimum possible private value u = 29, which is equal to p 1 u = 1. However, since the conditional execution probability of a limit sell order placed at p 2 = 31 is substantial, i.e., s, = , the trader with u = 29 will realize a larger expected payoff of (31 29) = from placing a limit sell order at p 2. Therefore, she will prefer to submit a limit sell order at p 2 rather than submit a market sell order at p 1. The same comparison also applies to all traders with u [29, 30]. However, we can show that, ceteris paribus, if the trader s private value distribution has a wider support with [27, 36], submitting a market sell order at p 1 in state ω 51 becomes optimal for all traders with u [27, ]. 17 Note that in state ω 51, the conditional execution probability (0.8138) of a limit sell order placed at p 2 = 31 is much higher than that (0.2025) of a limit sell order placed at p 3 = 32. First, a sell order p 2 has price priority over a sell order at p 3. Second, due to time priority, a limit sell order placed at p 3 in state ω 51 can only be executed after the execution or expiration of the existing limit sell order (with age 1) at p 3. Hence, the range of private values, [29, ], in which it is optimal to submit a limit sell order at p 2 is substantially wider than the range of private values, ( , ], in which it is optimal to submit a limit sell order at p 3.